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Journal of Electronic Imaging, 3(3), pages 295-317, 1994.

Spectral Analysis and Minimization of Moiré Patterns in Colour Separation

Isaac Amidror,

Roger D. Hersch,

Victor Ostromoukhov

Swiss Federal Institute of Technology (EPFL),

Peripheral Systems Laboratory (LSP),

1015 Lausanne, SWITZERLAND.

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Abstract

Undesired moiré patterns may appear in colour printing for various reasons. One of the

most important reasons is interference between the superposed halftone screens of the

different primary colours, due to an improper alignment of their frequencies or orientations.

In this article we explain the superposition moiré phenomenon using a spectral model

which is based on Fourier analysis. After examining the basic case of cosinusoidal grating

superpositions we advance, step by step, through the cases of binary gratings, square grids

and dot screens, and discuss the implications on moirés between halftone screens in colour

separation. Then, based on these results, we focus on the moiré phenomenon from a

different angle, the dynamic point of view: we introduce the moiré parameter space, and

show how changes in the parameters of the superposed layers vary the moiré patterns in the

superposition. This leads us to an algorithm for moiré minimization which provides stable

moiré-free screen combinations for colour separation.

Keywords: Moiré effect, moiré minimization, colour printing, halftoning, Fourier

analysis.

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1. Introduction

The moiré phenomenon is an optical effect which appears in some circumstanceswhen periodic or quasiperiodic structures (such as line gratings, dot screens, etc.) areintersected. It consists of a visible pattern which is clearly observed at the intersection,although it does not appear in any of the original structures [1]. While this phenomenonhas useful applications in several fields, such as in strain analysis or in the detection andmeasurement of slight deflections or deformations [2, 3, 4], in other situations moirépatterns may have an unwanted, adverse effect. Such is the case in the field of colourseparation: moiré patterns which appear in some circumstances between the dot screensused for colour printing may severely deteriorate the quality of the resulting image. It istherefore important to understand the nature and the causes of these moiré patterns inorder to know how to avoid, or at least minimize, their adverse effect on colourprinting.

Unwanted moiré patterns may appear in the printing process for several possiblereasons. For example, if the original image itself is already halftoned, or contains someother periodic fine details, then a moiré pattern may be caused as an interferencebetween the periodic fine details of the original input image and either the inputscanning frequency or the printing halftoning frequency. This type of moiré mayalready appear in black and white printing; but in the field of colour printing, by far themost (in)famous moiré problem is due to the superposition of the halftone screens of thedifferent process colours. This last type of moiré phenomenon is the subject of thepresent article.

The moiré effect between intersected structures occurs because of the geometricdistribution of dark and bright areas in the superposed image: areas in which the darkelements of the original structures cross each other contain less colorant than areaswhere the original structure elements fall between each other and fill the white spacesbetter.

Several mathematical approaches have been used to explore the moiré phenomenon.The geometric model [4, 5, 6], which is based on the geometric properties of thesuperposed layers (their periods and angles) leads to formulas that can predict, undercertain limitations, the geometric properties of the moiré patterns. Another approach isthe indicial representation of sets of curves [7]; this is a pure algebraic approach, whichalso yields the same formulas.

For a more founded treatment of the moiré phenomenon, however, an alternativemodel is required, based on spectral analysis in the frequency domain rather than on ageometric or algebraic analysis in the image domain. This spectral model has several

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advantages over the simple period-based geometric or algebraic approaches: (1) it lendsitself easily to a simple, illustrative graphic representation (the vector diagram); (2)using the Fourier transform theory it introduces the possibility of exploring thebidimensional spectrum of the superposed image from which not only the geometricproperties of the moiré pattern (angle and period) can be found, as in the otherapproaches, but also the wave form (profile) of the moiré and its strength (perceptualcontrast); (3) this approach affords a better understanding of the more complex moirécases and provides an easy explanation of multiple grating moirés, where the geometricanalysis may become too complicated. All the above points and terms will be explainedand illustrated below.

In this article we will deal basically with the moiré cases which lead to superpositionmoirés in colour separation. The explanation of this phenomenon is presented here stepby step, in a systematic way: we start with the simplest case, the intersection ofcosinusoidal gratings, and gradually proceed through the cases of binary gratings andsquare grids to the intersection of dot screens — which is the basis for the moiréphenomenon in colour separation. This is done based on a useful notational system thatwe introduce for the identification, classification and labelling of the moiré effects; thisformalism is also used for the definition of fundamental terms such as the order of amoiré, singular moiré states, etc. Then, based on these results, we focus in the secondpart of the article on the moiré phenomenon from a different angle — the dynamic pointof view: we introduce the moiré parameter space, and show how changes in theparameters of the superposed layers vary the moiré patterns in the superposition. Thisleads us to an algorithm for moiré minimization which provides stable moiré-freescreen combinations for colour separation. We present the algorithm, and conclude by adiscussion of the results obtained.

2. The spectral model – background and basic notions

The spectral model is based on the duality between the original image in the imagedomain and its spectrum in the frequency domain, through the Fourier transformation.By analyzing properties both in the original image and in its spectral representation thisapproach can benefit from the advantages of both domains.

In this article we are mainly interested in the bidimensional (2D) case, i.e. in 2Dimages in the (x,y) plane and their 2D spectra in the (u,v) plane, which are obtained bythe 2D Fourier transform. The following paragraphs list the basic properties of theimage types we are concerned with and review the implications of these properties bothin the image and in the spectral domains.

First, it should be noted that we will only deal here with moiré effects in themonochromatic case; moirés in colour separation will be simulated by the superposition

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of the 3 or 4 black-and-white separation films, without taking the ink colours intoconsideration. In the monochromatic case each printed image can be represented in theimage domain by a reflectance function, which assigns to any point (x,y) of the image avalue between 0 and 1 representing its light reflectance: 0 for black (i.e. no reflectedlight), 1 for white (i.e. full light reflectance), and intermediate values for in-betweenshades. Since the superposition of black and any other shade always gives black, thissuggests a multiplicative model for the superposition of monochromatic images. Thus,when N monochromatic images are superposed, the reflectance of the resulting image isgiven by the product of the reflectance functions of the individual images:

r(x,y) = r 1(x,y) r 2(x,y) ... rN(x,y) (1)

According to the convolution theorem [9, p. 244; 8, p. 18] the Fourier transform of theproduct function is the convolution of the Fourier transforms of the individualfunctions. Therefore if we denote the Fourier transform of each function by therespective capital letter and the 2D convolution by **, we get:

R(u,v) = R1(u,v) ** R2(u,v) ** ... ** RN(u,v) (2)

Second, here we are basically interested in periodic images, such as gratings orscreens, and their superpositions. This implies that the image spectrum is not acontinuous one but rather consists of discrete impulses, corresponding to thefrequencies which appear in the Fourier series decomposition of the image [9, p. 204].A strong impulse in the spectrum indicates a pronounced periodic component in theoriginal image at the frequency and direction of that impulse.

Each impulse in the 2D spectrum is characterized by two properties: its geometriclocation (or impulse location), and its amplitude (see Fig. 1). To the geometric locationof any impulse is attached a frequency vector f in the spectrum plane, which connectsthe spectrum origin with the geometric location of the impulse. This vector can beexpressed either by its polar coordinates (f,α), where α is the direction of the impulse

and f is its distance from the origin (i.e. its frequency in that direction); or by itsCartesian coordinates (fu, fv), where fu and fv are the horizontal and vertical components

of the frequency. In terms of the original image, the geometric location of an impulse inthe spectrum determines the frequency f and the direction α of the corresponding

periodic component in the image, and the amplitude of the impulse represents theintensity of that periodic component in the image.

In addition to the periodicity of the original images we assume, unless otherwisementioned, that each of the original images is regular in the sense that its components(lines or dots) are identical, parallel and equidistant.

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The periodic images with which we are dealing here are basically of a symmetricnature (gratings, screens, etc.). For the sake of simplicity, we also assume that the givenimages are not shifted, but indeed centered symmetrically about the origin. As a result,we will deal with images (and image superpositions) which are real and symmetric, andwhose spectra are consequently also real and symmetric [9, pp. 14-15]. This means thateach impulse in the spectrum (except for the DC at the origin) is always accompaniedby a twin impulse of an identical amplitude, which is symmetrically located at the otherside of the origin as in Fig. 1 (their frequency vectors are f and –f). The complexspectra for the case of non-centered images can be readily obtained whenever requiredby applying the shift theorem [9, pp. 104-107, 244].

Another important property of our images comes from the fact that most printingdevices are only bilevel, i.e. they are only capable of printing solid ink or leaving thepaper unprinted, but they cannot produce intermediate ink tones. (This is also true formost colour printing devices, where each of the primary colours is bilevel). In suchdevices the visual impression of intermediate tone levels is usually obtained by meansof the halftoning technique, i.e. by breaking the continuous-tone image into small dotswhose size depends on the tone level. Therefore, in most practical cases the reflectancefunction of a printed image is binary, only taking values 0 and 1 (signifying theexistence or absence of ink on the white paper, respectively).1 For colour separation wewill basically be interested in images consisting of binary dot-screens and theircombinations; but before we get there we will first study the basic case of cosinusoidalcontinuous-tone gratings, and then we will proceed through binary gratings and theirintersections to the case of binary dot screens.

3. Superposition of 2 cosinusoidal gratings

Let us first look at the case of gratings with a cosinusoidal wave form. Sincereflectance functions always take values between 0 and 1, the cosinusoidal reflectancefunction has the form (see Fig. 2(a)):

r 1(x,y) = 0.5 cos (2πf1x) + 0.5 (3)

This periodic function has a frequency of f1 cycles per unit, i.e. its period is T1=1/f1

units (in the x direction). Similarly, the reflectance function of a cosinusoidal gratingwith a frequency of f2 which is rotated by an angle α (Fig. 2(b)) is given by:

1 It should be noted that in reality, looking under a microscope, the border between black and white areas in animage does not appear to be a sharp transition, but rather a gradual transition in the distribution of the microscopicink particles. This microscopic view should not, however, invalidate the binary property assumed in the macroscopicview of the image.

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r 2(x,y) = 0.5 cos (2πf2[xcosα + ysinα]) + 0.5 (4)

The 2D Fourier transform of each of these reflectance functions consists of exactly 3impulses (Fig. 2(d), (e)). In fact, it is the sum of the Fourier transform of thecosinusoidal term which consists of 2 symmetric impulses of amplitude 1/4, located at adistance of fi=1/Ti from the origin at the respective angle; plus the Fourier transform of

the additional constant 0.5, which is an impulse of amplitude 1/2 at the origin (the DCimpulse).

According to Eqs. (1) and (2), the spectrum of the superposition of r 1(x,y) and r2(x,y)is the Fourier transform of their product, i.e. the convolution of their individual spectraR1(u,v) and R2(u,v). Since in our cosinusoidal case each of these spectra consists of 3impulses, their convolution consists of 9 impulses (see Fig. 2(f)). Using the graphicalmethod of “move and multiply” [9, pp. 29-30; 8, pp. 13-14] we see that the convolutionis 0 throughout the (u,v) plane, except at the points where the impulses of the movingcopy of R2(u,v) fall whenever the origin of R2(u,v) is placed on top of an impulse ofR1(u,v). This means that the geometric location of the impulses in the convolution canbe found simply by placing on top of each impulse of R1(u,v) a centered copy of R2(u,v)(or vice versa, since convolution is commutative). The amplitude of each impulse thusreceived is the product of the amplitudes of the 2 impulses involved: the impulse in thefirst spectrum on top of which the moving spectrum is centered, and the impulse in themoving spectrum which then defines the location of the impulse in question. If a newlygenerated impulse falls on top of an already existing impulse, their amplitudes aresummed up. The amplitudes received in our case are 1/4 for the DC impulse, 1/8 for the2 impulse pairs of the original cosines, and 1/16 for the 2 new impulse pairs generatedby the convolution (see bottom row in Fig. 2).

The spectrum resulting from the convolution contains all the impulse pairs of theoriginal spectra (only their amplitudes have been modified, but not their geometriclocations). However, 2 new pairs of impulses which did not exist in any of the originalspectra have appeared in the convolution (see Fig. 2(f)). The geometric locations ofthese new impulse pairs are determined by the vectorial sum and the vectorialdifference of the frequency vectors of the original impulse pairs, namely: f 1+f 2 , –f 1–f2,and f1–f2, f2–f1.

Since each impulse pair in the spectrum reflects a periodic component with thecorresponding frequency and angle in the original image, these 2 new impulse pairssuggest that the superposition of the 2 original images includes 2 new periodiccomponents which did not exist in either of the original images. And indeed, looking atthe superposed image (Fig. 2(c)) we can identify these new periodic components: Themore obvious one has the frequency and direction of the difference vector, f 1–f2; but theother one, with the frequency and the direction of the sum vector, f1+f 2, can also beidentified. The first periodic component is more visible than the other since its

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frequency is lower, i.e. its period is larger. While the frequency of the vector sum isalways larger than the frequencies of each of the individual vectors, the frequency ofthe vector difference may be significantly smaller than either of the originalfrequencies. Consequently, the periodic component in the superposed image whichcorresponds to the vector difference can have a significantly larger period, and thereforebe much more visible, than the cosines of the original images. This prominent periodiccomponent is, in fact, the moiré effect seen in the superposition of the 2 original images(Fig. 2(c)).

The fact that the eye cannot distinguish fine details above a certain frequency (i.e.below a certain period) suggests that the human visual system model includes a low-pass filtering stage. This is a bidimensional bell-shaped filter whose form is anisotropic(since it appears that the eye is less sensitive to small details in diagonal directions suchas 45° [10, p. 531]). However, for the sake of simplicity this low-pass filter can beapproximated by the visibility circle, a circular step function around the spectrum originwhose radius represents the cutoff frequency (i.e. the threshold frequency beyond whichfine detail is no longer detected by the eye). Obviously, its exact radius depends onseveral factors such as the contrast of the observed details, the viewing distance, lightconditions, etc. If the frequencies of the original image elements are beyond the borderof the visibility circle in the spectrum, the eye can no longer see them; but if thefrequency of the vectorial difference falls inside the visibility circle, then a moiré effectappears in the superposed image. (In fact, the visibility circle has a hole in its center,since very low frequencies cannot be seen, either).

4. Superposition of three or more cosinusoidal gratings; singularmoiré states

If we now superpose a third cosinusoidal grating on top of the first two, the resultingspectrum will be the convolution of all three spectra — i.e. the result of convolving thethird spectrum with the convolution of the first two spectra. The geometric location andthe amplitude of the impulses in the resulting spectrum can be found graphically in thesame manner as above: a centered copy of the new 3-impulse spectrum of the thirdcosinusoidal grating is placed on top of each of the 9 impulses of the 2-layerconvolution, thus generating 9 additional impulse pairs in the combined spectrum (Fig.3). The amplitude of each of the impulses of the copied spectrum is scaled by theamplitude of the impulse on top of which it has been copied. If any of the newlygenerated impulses fall inside the visibility circle, a new periodic component (or moiréeffect) can be seen in the image.

Generalizing this to the superposition of N cosinusoidal gratings, we see that the finalconvolution contains the frequency vectors of each of the original images as well as allthe frequency vectors obtained in each successive convolution. This means that the final

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convolution contains all the frequency vectors which can be obtained as a vectorial sumof 1, 2, ... or N frequency vectors, one (or none) from each original spectrum. If weconsider the DC impulse of each spectrum as having a zero frequency vector, it can besaid that each of the individual spectra contributes one of its frequency vectors to everyvectorial sum. In other words, the frequency vector f of any individual impulse in thefinal convolution (i.e. in the spectrum of the superposition of the N images) is avectorial sum of N frequency vectors f ' i, where f' i is one of the 3 frequency vectors

contained in the spectrum of the i-th image:

f = f '1 + f '2 + ... + f 'N(5)

If θi are the angles that the N original cosinusoidal images form with the positivehorizontal axis and fi are their frequencies, then the two Cartesian components of the

above vectorial sum can be written as:fu = f1 cosθ1 + f2 cosθ2 + ... + fN cosθN

fv = f1 sinθ1 + f2 sinθ2 + ... + fN sinθN

and the frequency, the period and the angle of the impulse in question are given by thelength and the direction of the sum vector f:

f = f u2+ f v

2

T = 1/ fϕ = arctan(fv / fu) (6)

The way in which any frequency vector f in the spectrum of the superposition isobtained from frequency vectors fi in the individual spectra of the original images can

be shown graphically using the geometric rules of vector addition, as shown in Fig. 10.Such vector diagrams can be drawn for any frequency vector f in the spectrum of thesuperposed image, but they are of particular interest for those impulses in the spectrumwhich correspond to moirés in the superposed image. The vector diagrams provide aclear illustrative explanation of the nature of any moiré in question.

If the impulse whose frequency vector is f falls inside the visibility circle andrepresents a visible moiré in the superposition of the N original images, the aboveformulas (6) in fact express the frequency, the period and the angle of this moiré. Notethat in the special case of N=2, where a moiré effect occurs due to the vectorial sum ofthe frequency vectors f 1 and –f 2, these formulas are reduced to the familiar

geometrically obtained formula of the moiré effect between 2 layers [4]:

T = T1T2

T12 + T2

2 – 2T1T2 cosα(6.1)

(where T1 and T2 are the periods of the 2 original images and α is the angle differencebetween them, θ1–θ2).

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Eqs. (5) and (6) only describe the geometric properties of an impulse in the spectrumof the superposition (and of the periodic component or moiré which it represents in theimage domain). The amplitude of any individual impulse, which represents the strengthof the corresponding periodic component in the image, is a product of the amplitudes ofthe N impulses from which it has been received in the convolution, one from each of theN spectra:

a = a1a2 ... aN (7)

Note however that if 2 or more impulses in the convolution happen to fall on top ofeach other in the same location, their individual amplitudes are summed up.

As can be seen from Eqs. (5) and (7), the spectrum convolution can be seen as anoperation in which frequency vectors of the original spectra are added vectorially,whereas the corresponding impulse amplitudes are multiplied. Note that since all theconvolved spectra are real and symmetric about the origin (see Sec. 2), the resultingspectrum is also real and symmetric, and contains for each impulse at location f anidentical twin impulse at –f.

We have seen that if one or several of the new impulse pairs in the convolution fallclose to the origin, inside the visibility circle, they imply the existence of one or severalmoirés with significant periods in the superposed image (Fig. 3(b), (d)). An interestingspecial case occurs when some of the impulses of the convolution fall exactly on top ofthe DC impulse at the origin. This happens for instance in the superposition of 2identical gratings at the same angle, or when 3 identical gratings are superposed atangle differences of 120° between each other. As can be seen from the vector diagrams,these are limit cases in which the vectorial sum of the frequency vectors is exactly 0.This means that the moiré frequency is 0 (i.e. its period is infinitely large), and thereforethe moiré is not visible. This situation is called a singular moiré state; but although themoiré effect in a singular state is not visible, this is a very unstable moiré-free statesince any slight deviation in the angle or frequency of any of the superposed imagesmay cause the new impulses in the spectrum to move slightly from the origin, thusgenerating a moiré effect with a very significant period [6, Figs. 6,8].

5. Binary square waves and their spectra

Let r(x) be a one dimensional binary (0, 1 valued) periodic square wave. We willdenote the period of this function by T and the width of its white square pulse (theopening) by τ (see Fig. 4). As usual, we assume that the square wave is symmetrically

centered around the origin, so that both the original image and its spectrum are real andsymmetric.

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A periodic square wave can be expressed by means of its Fourier series expansion, asan infinite series of weighted sine and cosine functions, at the fundamental frequency of1/T and all its harmonics. The general expansion (or decomposition) of a onedimensional periodic function into a Fourier series is given by:

r(x) = a0 + 2∞

∑n=1

an cos(2πnx/T) + 2∞

∑n=1

bn sin(2πnx/T) (8)

[9, p. 205], where the Fourier series coefficients are given by:

an = 1T

r(x) cos(2πnx/T) dx0

T

bn = 1T

r(x) sin(2πnx/T) dx0

T

(the integration may be done over any 1-period interval, such as -T/2...T/2, etc). In thespecial case of a symmetric binary square wave, where r(x)=1 between -τ/2...τ/2 and 0

elsewhere (see Fig. 4), these coefficients are:a0 = τ/T

an = (1/nπ) sin(nπτ/T) = (τ/T) sinc(nτ/T)(9) bn = 0

(there are no sine components here, due to the symmetry of the square wave). The factthat the square wave can be expressed as a constant a0 plus an infinite sum of cosine

functions, implies that the Fourier transform of the square wave contains a DC impulsewhose amplitude is a0, plus an infinite series of impulse pairs located at the frequencies

of ±n/T. Since the n-th impulse pair is the Fourier transform of the n-th cosine in theFourier series decomposition of the square wave (the n-th harmonic), its amplitude isgiven by the n-th cosine coefficient, an (Eq. 9). The spectrum of the square wave is

therefore:

R(u) = ∞

∑n=–∞

an δ(u–n/T) (10)

(where δ(u) is the impulse symbol). This is an impulse train (or a “Dirac comb”)

which samples the continuous function (“envelope”) g(u)=(τ/T)sinc(τu) at the

fundamental frequency u=1/T and all its harmonics, u=n/T (Fig. 4). The amplitude ofthe impulses oscillates and fades out symmetrically in both directions from the center.

Note that the period T of the square wave determines the interval 1/T between eachtwo successive impulses of the comb in the spectrum, while the opening τ (0 ≤ τ ≤ T)

determines the length of the lobes in the envelope sinc function. The height of theenvelope at the origin, i.e. the amplitude of the DC impulse, is determined by theopening ratio of the square wave, τ/T (0 ≤ τ/T ≤ 1).

An interesting observation from the formula of an (Eq. 9) is that when the opening

ratio τ/T of a given square wave is rational, i.e. if it can be expressed as l/k where l andk are integers, then for any n which is a multiple of k the impulse amplitude an is zero.

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For instance, if τ/T=1/2 then every even impulse in the comb has a 0 amplitude, and if

τ/T=1/4 or 3/4 then every fourth impulse in the comb is zero.

Let us mention here one more result that will be used later: If R(u) is the Fouriertransform of a symmetric square wave r(x), then the Fourier transform of the “negative”wave 1–r(x) is given by:

F(1–r(x)) = F(1) –F(r(x)) = δ(u) – R(u) = { 1–R(0) u=0–R(u) u≠0

(11)

Therefore, if an are the impulse amplitudes in the spectrum of r(x), then the impulseamplitudes in the spectrum of 1–r(x) are: c0 = 1–a0, cn = –an.

Proceeding now to the 2D case, the general expansion of a periodic function r(x,y)with periods Tx and Ty in the x and y directions into a 2D Fourier series is given by:

r(x,y) = ∑m = -∞

∞

∑n = -∞

∞

am,n cos2π(mx/Tx + ny/Ty) + ∑m = -∞

∞

∑n = -∞

∞

bm,n sin2π(mx/Tx + ny/Ty)

where:am,n = 1TxTy

0

Tx

r(x,y) cos2π(mx/Tx + ny/Ty) dx dy 0

Ty

bm,n = 1TxTy

0

Tx

r(x,y) sin2π(mx/Tx + ny/Ty) dx dy 0

Ty

If r(x,y) is symmetric, the infinite sum only contains cosine terms, each of which is afunction of both mx and ny. Each of these cosine terms, multiplied by a coefficient am,n,

gives (isolates) the contribution to the image r(x,y) of the cosinusoidal periodiccomponent in the direction and frequency of the (m,n)-th harmonic. The spectrum(Fourier transform) of r(x,y) is, therefore, an impulse nailbed which contains for each ofthe (m,n)-th harmonics an impulse of amplitude am,n. The frequency vector of each

impulse in the nailbed denotes the direction and the frequency of the correspondingperiodic component in the image. In the following section we will discuss the specialcase where r(x,y) is a binary square wave. Note that since a 2D square wave is basicallya constant extension of a 1D square wave into the perpendicular direction, its 2DFourier series representation (and therefore its 2D spectrum) are in fact of 1D nature.

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6. Superposition of binary gratings; higher order moirés; thenotational system

The reflectance function of a binary grating is a bidimensional binary periodic squarewave; for the sake of convenience we will assume it is also symmetric. The reflectancefunction of a binary grating with period T1 and opening τ1 in direction x (Fig. 5(a)) is

given by the Fourier series:

r1(x,y) = a 0(1) + 2

∞

∑n=1

a n(1) cos(2πnx/T1) (12)

where the coefficients a(1)0 and a(1)

n are similar to a0 and an in Eq. (9) above, with only

T1 and τ1 replacing T and τ. Similarly, the reflectance function of a binary grating with

period T2 and opening τ2, which is rotated by angle α (Fig. 5(b)), is given by the Fourier

series:

r2(x,y) = a 0(2) + 2

∞

∑n=1

a n(2) cos(2πn[xcosα + ysinα]/T2) (13)

where a(2)0 and a(2)

n are similar coefficients with T2 and τ2.

The Fourier transform R1(u,v) of the reflectance function r1(x,y) is a symmetric 1Dimpulse train on the u axis (Fig. 5(d)), whose envelope is g(u) = (τ1/T1)sinc(τ1u). Theintervals between the impulses are 1/T1 and their amplitudes are a(1)

n. Similarly, the

Fourier transform R2(u,v) of the reflectance function r 2(x,y) is a 1D symmetric impulsetrain with an envelope shape of g(u) = (τ2/T2)sinc(τ2u) on a line in the u,v plane which

has been rotated by angle α , with an impulse interval of 1/T2 and an impulse amplitudeof a(2)

n (Fig. 5(e)).

Let us now consider the superposition of the 2 binary gratings, r(x,y) = r 1(x,y)r 2(x,y).According to the convolution theorem (Eq. 2) its spectrum R(u,v) is the convolution ofspectra R1(u,v) and R2(u,v). This convolution can be carried out graphically by the“move and multiply” method as in the case of 2 cosinusoidal functions, the onlydifference being that now each of the individual spectra contains an infinite number ofimpulses (a “comb”) rather than only 3 impulses. The result of the convolution, in termsof impulse locations, is an infinite oblique lattice of points (frequency vectors) whichresults from placing a centered, parallel copy of the comb R2(u,v) on top of eachimpulse of the comb R1(u,v), or vice versa (see Fig. 5(f), and compare with thecosinusoidal case shown in Fig. 2(f)). More precisely, the results of the convolution canbe formulated as follows:

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(a) The frequency vectors of the impulses in the spectrum of the superposed gratings arethe vectorial sums of each frequency vector on the comb of the first grating and eachfrequency vector on the comb of the other grating. All these frequency vectors (orrather their corresponding impulse locations) form the above-mentioned lattice.

(b) The amplitude of the (m,n)-th impulse on this oblique lattice is the product of theamplitudes of the m-th impulse in the first comb and the n-th impulse in the secondcomb:

am,n = a(1)m a(2)

n

= (1/ π2mn) sin(πmτ1/T1) sin(πnτ2/T2)

(14)= (τ1τ2/T1T2) sinc(mτ1/T1) sinc(nτ2/T2)

As in the cosinusoidal case, we see that here too the superposition of gratingsintroduces new impulses in the spectrum. If any of these impulses fall inside thevisibility circle, as in Fig. 5(f), it indicates that in the superposed image there is a visibleperiodic component (i.e. a moiré effect) at the corresponding direction and frequency.

Results (a) and (b) above lead to the following interesting observations:

(1) The angle difference between the two gratings only influences the geometriclocation of the impulses, but not their amplitudes. In the image domain this means thatthe angle difference between the two gratings only influences the angle and period ofeach moiré, but not its amplitude. (In case of more than 2 superposed gratings, however,angle changes may cause some impulses to fall on top of each other, in which case theiramplitudes are summed up).

(2) Varying the opening τ of any of the gratings while the period T remains fixed only

influences the amplitude of the impulses, but not their location. In the image domainthis means that only the amplitude of the moiré is influenced, but not its angle or period.

The generalization of the results obtained above to the superposition of N binarygratings is straightforward. In fact, the geometric location of each impulse in theresulting spectrum is determined by the vectorial sum of frequency vectors, one fromeach of the superposed gratings, while the amplitude of the individual impulses is theproduct of the amplitudes of the involved impulses. This is very similar to the case ofcosinusoidal gratings, except that the spectrum of a binary grating consists of an infinitecomb of impulses rather than only 3 impulses, and therefore in the case of binarygratings the convolution of N spectra gives an infinite nailbed of impulses rather thanjust a finite number (3N) of impulses. This means that in the case of binary gratings eachof the components f ' i in Eq. (5) may come from any impulse of the comb of the i-thspectrum. In fact, if f i is the frequency vector of the fundamental impulse in the i-thspectrum, then the frequency vector of its ki-th harmonic impulse is kifi, and Eqs. (5)and (7) for the general (k1,k2, ... ,kN)-impulse in the convolution become:

15

f k1,k2,...,kN = k1f 1 + k2f 2 + ... + kNfN

(15)

ak1,k2,...,kN = a(1)

k1 a(2)

k2 ... a(N)

kN

(16) = 1

πNk1k2...kN

sin (πk1τ1/T1) ... sin (πkNτN/TN)

= τ1τ2...τNT1T2...TN

sinc (k1τ1/T1) ... sinc (kNτN/TN)

where ki are integer numbers. If 2 or more impulses in the convolution happen to fall

on the same point (i.e. they have identical frequency vectors), their individualamplitudes are summed up.

The Cartesian components of the vectorial sum in (15) are:

fu k1,k2,...,kN = k1f1 cosθ1 + k2f2 cosθ2 + ... + kNfN cosθN

fv k1,k2,...,kN = k1f1 sinθ1 + k2f2 sinθ2 + ... + kNfN sinθN

(

17)

and as in the cosinusoidal case they can be inserted into Eq. (6) in order to obtain thefrequency, the period and the angle of the impulse in question (and of the moiré it mayrepresent, in case it falls inside the visibility circle).

As a result, two main differences emerge between the superpositions of cosinusoidalor binary gratings: first, in the case of binary gratings, each impulse in the visibilitycircle (like any other impulse in the spectrum) belongs to an infinite comb of impulses,in the same direction, through the origin (see Fig. 5(f)). This means that each moiré isrepresented in the spectrum by an infinite series of impulses. The fundamental impulse(i.e. the first impulse next to the DC) in this comb determines the period and thedirection of the moiré, and therefore it is still sometimes called, as in the cosinusoidalcase, “the impulse of the moiré”. If further harmonic impulses of this comb also fallinside the visibility circle, the profile shape of the moiré is no longer perceived as apure cosinusoidal function, but rather as a more complex form (a sum of cosines).

Second, in the case of binary grating superposition, the visibility circle may alsocontain impulses which originate from higher harmonic impulses in the combs of theindividual spectra. This means that, unlike in the cosinusoidal case, moiré effectsbetween binary gratings can also be obtained from higher harmonics of the fundamentalgrating frequencies. Such moiré effects are called higher order moirés [11]. This isdemonstrated for the case of 2 superposed gratings in Fig. 6; note that the visible moiréeffect is caused here by the vectorial sum (or rather difference) of f 1, the fundamentalfrequency of the first grating, and twice f2, i.e. the second harmonic of the other grating,while the vectorial difference f 1–f2 is outside the visibility circle. In the image domain

16

this means that the visible moiré is actually due to the intersection of every second linein the first grating with each line of the second grating. If both f1–f2 and f 1–2f2 were

inside the visibility circle, then both the first and the second order moirés could beobserved simultaneously in Fig. 6(c). Fig. 7 shows some of the various order moiréswhich may exist between 2 superposed gratings.

It will be useful to present at this point the systematic notational formalism that weintroduce for the following discussions, which provides an unambiguous means for theidentification, classification and labelling of the moiré effects. We will call the N-grating moiré whose fundamental impulse is the (k1,k2, ... ,kN)-impulse in the spectrumconvolution (see above) a (k1,k2, ... ,kN)-moiré; its comb contains the convolutionimpulses (nk1,nk2, ... ,nkN) for any integer n (this can be seen for N=2 in Figs. 5, 6).This moiré is singular if ∑kifi = 0. Note that sometimes, when no confusion may occur,

it is convenient to use a shorthand notation in which zero indices and negative signs areomitted. For instance, the (2,0,-1,1)-moiré between 4 gratings may be simply called a{2,1,1}-moiré, and both the 4 grating (1,0,-1,0)-moiré and the 2 grating (1,-1)-moirémay be called in short {1,1}-moirés. This shorthand notation is sometimes moreconvenient than enumerating all the possible similar variants. The highest absolutevalue in the list is called the order of the moiré.

7. The profile of the moiré and its perceptual contrast

As has been shown above, each moiré in the superposition of binary gratings isrepresented in the spectrum by an infinite series of impulses (a Dirac comb). Theamplitudes of these impulses are the coefficients of the Fourier series development ofthe function which represents the profile of the moiré (i.e. the intensity of the moiré atany point along a cross-section of the image in the moiré direction). The exact profileshape of the moiré in the image domain can be reconstructed as an infinite sum ofcosines, by inserting the amplitudes an=ank1,nk2,...,nkN

of the moiré comb impulses given

by Eq. (16) into Eq. (8) (with bn=0 for any n, since we are dealing with symmetric

cases). An approximate profile shape of the moiré can be obtained from the first fewimpulse pairs of the comb which fall inside the visibility circle, i.e. as a sum in theimage domain of only the first few cosine terms in the Fourier series (8) (plus a constantterm due to the DC impulse, which represents the average level of the profile). Acoarser, cosinusoidal shaped approximation of the moiré profile can already be obtainedfrom the DC and the fundamental impulse pair alone (whose frequency vector gives thebasic angle and frequency of the moiré).

As we have seen, our mathematical model assigns to each point of the image areflectance value between 0 and 1, where 0 means black, 1 means white, andintermediate values represent in-between reflectance values. This applies, in particular,to the moiré profiles and the image projections which are calculated as described above

17

based on this model.2 Note that although the original images and their superpositionsare all binary and only take values 0 and 1, their moiré profiles may also haveintermediate values. In fact, the value of each point on a profile represents the averageratio of white per unit of length (or area), i.e. the average reflectance at that point. Forexample, in the (1,-1)-moiré between 2 identical gratings where the width of the blackand white lines is identical (i.e. the opening ratio is τ/T=0.5; see Fig. 8(b)), then the

value of the moiré profile along the center of the dark moiré bands is 0 (no white at all),and this value gradually climbs up to 0.5 at the center of the bright moiré bands (whereblack and white are equally distributed, and therefore the white ratio is 0.5). This isshown graphically in the reflectance profile of the moiré, in Fig. 8(e); similar profilesare shown in Figs. 8(d),(f) for other opening ratios corresponding to the moirés of Figs.8(a),(c).

However, the difference between the maximum and the minimum reflectance of themoiré profile does not represent correctly the degree of contrast as it is perceived by thehuman eye. For example, in the (1,-1)-moiré between 2 identical gratings, the differencebetween the maximum and the minimum reflectance profile values are identical forgratings with opening ratios of τ/T=0.75 and τ/T=0.25, while the eye clearly sees a

much higher contrast in the second (darker) case (compare Figs. 8(a),(c) with theirreflectance profiles 8(d),(f)).3 The reason for this phenomenon is that the response (orsensibility) of the human visual system to light intensity is not linear in its nature, butrather close to logarithmic [12, pp. 27-29]. If we plot the intensities or the moiréprofiles logarithmically, i.e. in terms of density rather than in terms of reflectance, weget a more realistic representation of the perceptual contrast of the moiré which bettercorresponds to the human perception (see Fig. 8(g)–(i)). A still better correspondencecan be achieved by replacing the logarithmic approximation of the human visualresponse by an empiric function based on the experimental data obtained fromphysiological research [13, pp. 62–69].

8. Square grids and their superpositions

A square grid is in fact an orthogonal superposition (multiplication) of two identicalbinary gratings, which together form a pattern of identical squares. Therefore thespectrum of a square grid is the convolution of two identical but perpendicular combs,and its impulses are located on a square lattice. The amplitude of each impulse in the

2 The range of [0,1] is respected, of course, only by the precise profile functions which take into account

all the impulses till infinity. An approximation using only a finite number of terms, such as the DC plus

the first harmonic cosine (x2), may somewhat exceed the range of [0,1].3 Note that even the Michelson contrast [13 p. 34], defined as: (max–min)/(max+min), which is widely

used in literature, is not appropriate here — since for both cases 8(e),(f) it yields the same value, 1.

18

spectrum is given by Eq. (14); as we can see, the spectrum has the form of a nailbedwhich samples the continuous bidimensional function (“envelope”) g(u,v) =(τ/T)2sinc(τu)sinc(τv) at the points (u,v) = (m/T,n/T), where T and τ are the period and

the opening of both gratings. The shape of such a spectrum can be seen in Fig. 11(f).

The spectrum of a superposition of two square grids is the convolution of 2 suchnailbeds: a centered copy of one of the nailbeds is scaled and placed on top of eachimpulse of the other nailbed. But the superposition of 2 square grids can also be seen asa special case of 4 binary grating superperposition with two pairs of gratings, each pairconsisting of 2 identical but perpendicular gratings. It is clear, therefore, that eachimpulse pair in the spectrum of the superposition is accompanied by an identical,perpendicular impulse pair. This means that every moiré effect which occurs betweenany of the gratings involved always has a perpendicular twin; the visual effect in thesuperposed image therefore has a 90° symmetry (such as a square grid or screen).

The moiré effects which occur in the superposition of two square grids may originatefrom 2, 3, or from all of the 4 gratings involved. This means that the frequency vectorof the fundamental moiré impulse in the visibility circle of the spectrum may be avectorial sum of 2, 3, or 4 frequency vectors.4 Examples of these types of moirésbetween 2 square grids are shown in Fig. 9; their vector diagrams are shown in Fig. 10.In each case, the low frequency vectorial sum of the indicated frequency vectors fromthe 2 original spectra gives the fundamental frequency vector of the visible moiré effect,and therefore determines the angle and the frequency of the moiré. (Note that in order todetermine the exact profile shape of the moiré, its higher harmonic impulses are alsorequired, insofar as they are situated inside the visibility circle).

The generalization into the case of 3 or more superposed square grids isstraightforward. The general expressions for the frequency vector and the amplitude ofan impulse in the spectrum of the superposition are given by Eqs. (15, 16), where N isthe number of gratings involved, i.e. twice the number of the square grids.

9. Dot screens and their superpositions

A regular dot screen is a dot screen with perpendicular axes and equidistant, identicaldots. This is in fact a generalization of a square grid, since the square grid can beconsidered as a special case of a screen having square, white dots on black background(see Fig. 11(e)). A regular screen r(x,y) of white dots on a black background can be

4 A grating which does not explicitely participate in the generation of the moiré effect in fact contributes

its DC impulse: it contributes a zero frequency vector to the vectorial sum of the frequency vectors, and

the amplitude of its DC impulse to the product amplitude (see Eqs. 15, 16).

19

seen as a convolution of a “spot function” d(x,y) which describes a single white dot,with a nailbed of period T (see Fig. 11). Therefore, according to the convolutiontheorem, the spectrum of such a dot screen is the product of the continuous functionD(u,v) (the Fourier transform of the spot function), and the Fourier transform of thenailbed which is itself a scaled and stretched nailbed. This product is a nailbed whichsamples the “envelope” function D(u,v) at intervals of 1/T, scaling its amplitude by 1/T2

(see Fig. 11). In the case of a square white dot whose side is τ (as in Fig. 11), the spot

function is: d(x,y) = 2Π(x/τ,y/τ) (where 2Π(x/τ,y/τ) denotes the function whose value

is 1 within the square –0.5≤x,y≤0.5 and 0 elsewhere), and therefore the envelope of thespectrum is: D(u,v) = τ2 sinc(τu) sinc(τv) [9, p. 246]. In the case of a circular dot with

radius τ, the envelope D(u,v) is given by a Bessel function [9, p. 248], which is

sometimes called the sombrero function due to its circular symmetry.

As we can see, the shape of an individual dot d(x,y) of the screen determines theshape of the envelope function D(u,v) in the spectrum, and therefore it determines theamplitude of each of the impulses in the spectrum of the screen. However, thegeometric locations of the impulses in the spectrum of the screen are not influenced bythe dot shape, and they are determined only by the nailbed with which the dot is beingconvolved.

A regular screen of black dots on white background, which represents a halftonescreen with dot area coverage of less than 50%, can be seen as the “negative” of theabove white-on-black screen r(x,y), namely: 1 – r(x,y). Following Eq. (11), the Fouriertransform of 1–r(x,y) contains exactly the same impulses as R(u,v), the spectrum of thesquare grid r(x,y), where only the sign of their amplitudes have been inverted:cm,n = –am,n, and the DC impulse has an amplitude of: c0,0 = 1–a0,0. Here again, only the

impulse amplitudes have been influenced, but not their geometric locations.

The spectrum of the superposition of 2 dot screens is the convolution of the 2 bell-shaped nailbed spectra of the individual screens, as in the special case of square grids.Such a convolution process in the spectrum is repeated for each additional dot screenwhich is superposed in the image domain. If after the convolution the spectrum containsnew impulses inside the visibility circle which yield sufficiently strong perceptualcontrasts, then new visible periodic components with the respective angles andfrequencies appear in the superposed image as moiré effects. This is very similar to thespecial case of square grids discussed above (i.e. the case of square white dots on ablack background), and in fact the moiré effects received in both cases have verysimilar macroscopic properties (see Fig. 9). Note however that the microscopicproperties of the two cases may be quite different. The microscopic structure in thesuperposition of dot screens is distinguished by a fine pattern of small dot clusters,called rosettes [5], which can be seen even when no visible moirés are present.

20

The fact that the geometric location of the impulses in the spectrum of the screen doesnot change when the form of the individual dot is modified (or when the negative of thescreen is taken) is fundamental for the understanding of the moiré effects betweensuperposed screens. It means that the period and the direction of the moiré do notdepend on the shape of the individual dots of the superposed screen; changing the dotshape (including the dot size) may only affect the intensity and the profile of the moiréeffect. This is shown in Fig. 12.

10. The implications on moirés between halftone screens in colourseparat ion

A halftoned image with clustered screen elements is a screen of equidistant dots, inwhich the dot size is not constant but rather varies (according to the tone values in theoriginal image). The moirés between halftoned images are therefore based on the caseof regular dot screens which has already been described above. Since the colourseparation technique involves overprinting 3 or 4 halftoned images, one for each of theprimary colours [10, Sec. 26.6], undesired moiré patterns may occur between thesehalftone screens if special care is not taken.

In order to be clearly visible, a moiré between 2 or more screens should have a largeenough period (i.e. its fundamental impulse in the spectrum must be inside the visibilitycircle), and its perceptual contrast (see Sec. 7) should be relatively strong. However, itis important to note that in colour separation not only strong moirés with large periodsmay be harmful. Weaker moirés with large periods (say, 10 times the screen period ormore) can still be visible, as well as strong moirés with small periods (3-5 times thescreen period); the latter may cause a rough linen-like texture and give the printedimage a grainy aspect. As a rule of thumb it can be said that pronounced moiréstructures which are larger than 1 mm are already clearly visible in normal viewingconditions and therefore undesirable.

The minimization of moirés between superposed screens consists of findingparameter combinations for the superposed layers which give as far as possible moiré-free results. But the task of finding a good moiré-free combination of more than 2halftone screens is not an easy one, since many different moirés may appear at eachangle and frequency combination (this is illustrated in Figs. 7 and 13 for the case of 2superposed layers). Minimizing one of these moirés (by varying the screen frequenciesand angles) does not guarantee the minimization of the others, and in fact the task offinding a good screen combination is a trade-off between the contradictory tendenciesof the various moirés involved.

It should also be noted that not all the moiré minima are indeed stable solutions. Asdescribed in Sec. 4 many of the moiré-free combinations are in fact singular moiré

21

states; therefore any small deviation in the angle or frequency of any of the dot screensmay cause the reappearance of the moiré pattern in its full strength. Singular moiré-freestates may be used in colour separation only if high accuracy can be guaranteed for thescreen angles and frequencies; in fact, such is the case in the traditional screens withequal frequencies and equal angle distribution (30°-30°-30°). But moiré-freecombinations which are not singular should be preferable, since they are more stableand much less sensitive to small inaccuracies in the reproduction and printing process.

In halftoned images, different gray levels are obtained by variations in the size andsometimes also in the shape of the screen dots. But as we have seen, the envelope shapeand therefore the amplitude of each impulse in the nailbed spectrum of a dot screen arestrongly influenced by the shape of the dot, as well as by the opening ratio τ/T or the

dot size. This means that the intensity and the profile of the moirés between halftonescreens (but not their angles or periods!) are influenced by the shape and size of thedots, since they depend on the amplitude of the new impulses received in thesuperposition of these screens. In other words, even at a fixed frequency and anglescreen combination the intensity and the profile of the generated moiré (or moirés) maystill vary significantly (within their fixed periods and angles) as the halftone gray levelsare being changed. Furthermore, when several moirés are present simultaneously, eachof them may be dominant in a different combination of gray levels. These facts areclearly demonstrated in Fig. 13. It is therefore not sufficient to look for good screencombinations only at some predefined gray levels, since an innocent moiré in one graylevel may prove to be much stronger in another gray level. However, an exhaustive runthrough a large range of screen parameters and gray levels in order to find the bestscreen combinations proves to be impractical: due to the large amount of calculation,and also due to the fact that for most of the practically used halftone cells (such as blackcircles which gradually grow into squares at 50% and then gradually turn into whitecircles) we have no precise expressions for the impulse amplitudes. We thereforeselected a simpler approach for the moiré minimization, which is only based ongeometric parameters such as angles and frequencies (although perceptual intensities ofthe moirés are also taken implicitly into account by experimental measurements). Thisapproach will be explained in the following sections.

11. Dynamics of the moiré phenomenon; the moiré parameter space

In the foregoing sections we analyzed the moiré phenomenon between superposedlayers only from the static point of view. That is, given a fixed combination of anglesand frequencies for the individual layers we analyzed the effects of their superposition,both in the image domain and in the spectral domain (like in Figs. 5, 6, 9-10).

Based on these results, in the present section we will consider the dynamic aspects ofthe moiré phenomenon between superposed layers, i.e. how changes in the parameters

22

of the superposed layers affect the moiré phenomena in the superposition, both in theimage and in the spectral domains.5 Understanding these dynamic aspects will give us amore profound insight into the behaviour of the moiré phenomenon, and lead us to amethod for finding moiré-free screen combinations for colour separation.

Let us start with the simple case of two superposed regular dot screens. As we haveseen, any of the convolution impulses in the spectrum which fall within the visibilitycircle represent a moiré phenomenon between the two screens. Let the frequencies ofthe 2 screens be f1 and f2 and the angles they form with the positive horizontal axis be

θ1 and θ2. Then according to Eq. (17) the geometric location of the (k1,k2,k3,k4)-impulsein the spectrum is given by the Cartesian coordinates (fu k1,k2,k3,k4

, fv k1,k2,k3,k4) where:

fu k1,k2,k3,k4 = k1f1 cosθ1 + k2f1 cos(90°+θ1) + k3f2 cosθ2 + k4f2 cos(90°+θ2)

fv k1,k2,k3,k4 = k1f1 sinθ1 + k2f1 sin(90°+θ1) + k3f2 sinθ2 + k4f2 sin(90°+θ2)

(18)

or equivalently, in vector form:

fufv k1,k2,k3,k4

=

cosθ1 -sinθ1

sinθ1 cosθ1 k1

k2 f1

+

cosθ2 -sinθ2

sinθ2 cosθ2 k3

k4 f2

Obviously, if we now let the parameters fi and θi vary, the geometric location of the

(k1,k2,k3,k4)-impulse varies accordingly in the spectrum. For the sake of simplicity we

may assume without loss of generality that the first screen is fixed, with the angleθ1 = 0° and the frequency of f1, and only the second screen is free to vary. Thus, in the

superposition of 2 regular dot screens there are only 2 independent parameters (degreesof freedom); it is convenient to choose them to be the angle difference between the 2screens, α = θ2–θ1 = θ2, and their frequency ratio, q = f2/f1. Expression (18) for the

geometric location of the (k1,k2,k3,k4)-impulse in the spectrum can be rephrased,

therefore, as a function of the 2 variables α and q, as follows:

fu k1,k2,k3,k4 = k1f1 + q f1[k3 cosα + k4 cos(90°+α)]

fv k1,k2,k3,k4 = k2f1 + q f1[k3 sinα + k4 sin(90°+α)]

(19)

The distance of this impulse from the spectrum origin (i.e. the frequency of theimpulse) is given, according to Eq. (6), by:

5 We will restrict ourselves here only to dynamics of the geometric parameters, i.e. to the influence of

varying the angles and frequencies of the superposed layers on the resulting moirés. The effects of

varying the gray levels or the dot shapes in the superposed layers have already been discussed above (see

Figs. 8, 12, 13).

23

f k1,k2,k3,k4 = fu k1,k2,k3,k4

2 + fv k1,k2,k3,k4

2

and its period is: Tk1,k2,k3,k4 = 1/ f k1,k2,k3,k4

.

Let us take as an example the (1,1,-1,0)-moiré of Fig. 9(b), i.e. the impulse with k1=1,k2=1, k3=–1 and k4=0. As we can see in Fig. 10(b), the geometric location of this

impulse (denoted in the figure as the vectorial sum) is quite close to the spectrumorigin; and in fact, if we rotate and scale the free screen so that α and q vary towards

the values of α=45° and q=√2, the impulse will gradually approach the origin, and

finally reach it. At this precise point the period of the moiré is infinitely large, but atany values of α and q around this point the impulse in the spectrum will still be located

within the visibility circle, so that the moiré will have a large, visible period. It can besaid, therefore, that in the 2D parameter space defined by all the possible values of theparameters α and q, the point (α ,q)=(45°,1.4142) represents a singular state of the

(1,1,–1,0)-moiré. Moreover, this is the only singular point of the moiré, as a glance atFig. 10(b) shows: since the first screen is fixed, so is the sum of its two f 1 vectors; andonly at α=45°, q=√2 the vector –f2 of the second (varying) screen cancels it out and

brings the vectorial sum (the small arrow) to zero. A similar reasoning shows that any(k1,k2,k3,k4)-moiré between 2 screens has a single singular point in the 2D parameter

space.

However, every (k1,k2,k3,k4)-moiré may have a different singular point in the

parameter space. This is demonstrated in Fig. 14, which gives a panoramic view of theparameter space (α ,q) and illustrates the dynamics of the most important moirés (up to

the second order) which appear between the 2 superposed screens. Note that differentmoirés may have an identical singular point in the parameter space; this is the case, forinstance, in the (1,1,-1,0)- and (2,0,-1,1)-moirés (see Fig. 14). We will call such moirésisocentric since they share the same central (singular) point. The contour line aroundeach moiré in the figure delimits the parameter combinations (α ,q) for which the

fundamental impulse of the moiré is located inside the visibility circle.

Having illustrated the dynamics of moiré effects between two superposed dot screens,we are now ready to examine the case of 3 superposed screens, which is the basicconfiguration for colour separation.

For the sake of clarity, let us adopt the following notational conventions: we will callthe 3 dot screens, in descending order of their frequencies, the K-screen, the M-screenand the C-screen (shorthand for: black, magenta and cyan). We suppose without loss ofgenerality that the K-screen is fixed, with the angle θ1 = 0° and the frequency of fK, and

that only the C- and M-screens are free to vary. The angle between the K- and the M-screens will be denoted by α and the angle between the K- and the C-screens by β (see

Fig. 15).

24

The case of 3 superposed screens therefore has 4 independent parameters: the anglesα and β, and the frequency ratios qMK=fM/fK and qCK=fC/fK. The geometric location of the

(k1,k2,k3,k4,k5,k6)-impulse in the spectrum of the superposition can be expressed

(according to Eq. 17) as a function of these 4 parameters, as follows:

fu k1,...,k6 = k1fK + qMKfK [k3 cosα + k4 cos(90°+α)] + qCKfK [k5 cos(–β) + k6 cos(90°–β)]

fv k1,...,k6 = k2fK + qMK fK [k3 sinα + k4 sin(90°+α)]

+ qCKfK [k5 sin(–β) + k6 sin(90°–β)]

(20)

with fk1,...,k6 = fu k1,...,k6

2 + fv k1,...,k6

2 and Tk1,...,k6 = 1/ fk1,...,k6

.

Since the parameter space in this case is 4-dimensional, (α , β, qMK, qCK), a full graphic

description (like Fig. 14 for the 2D case) is no longer possible. But, except for theabstraction due to the 4 dimensions, the situation remains basically similar. The maindifference is that unlike in the 2D case, the locus of the singular points of a(k1,k2,k3,k4,k5,k6)-moiré in the 4D parameter space is no longer a single point, but rather

consists of a 2D manifold (a curved surface) within the 4D space. This is illustrated inFigs. 16-17.

As the parameters (α , β, qMK, qCK) vary away from this singular manifold, the period

of the moiré becomes smaller until at a certain distance from the singular manifold themoiré fades out and becomes practically invisible. If we “draw” in the 4D space thelocus of all the points (α , β, qMK, qCK) at which the (k1,k2,k3,k4,k5,k6)-impulse is locatedinside the visibility circle in the spectrum (i.e. f k1,...,k6

< visibility circle radius), we will get

around the singular manifold a 4D zone (which is analogous to the spherical zonearound each singular point in the 2D case of Fig. 14). We will call this zone theforbidden zone of the moiré in question. In 2D sections taken through the 4D parameterspace this forbidden zone may appear as a thick, curved or straight line (Fig. 17(a)), oras an elliptic or egg-like shape (Fig. 17(b)).

Now, if we “draw” in the 4D space the forbidden zones belonging to all the(k1,k2,k3,k4,k5,k6)-impulses in the spectrum which represent perceivable moirés, we will

get the following image: the singular manifolds of these moirés will appear dispersed inthe 4D space, each of them serving as a skeleton which is surrounded by the forbiddenzone of the corresponding moiré. These forbidden zones are sometimes intersected, andsometimes they only almost touch in the 4D space. The spaces left between theseforbidden zones include the parameter combinations for which no disturbing moiréoccurs between the 3 screens.

25

12. Finding moiré-free screen combinations for colour separation

The above discussion immediately suggests a practical way of searching for moiré-free combinations (or moiré minimization) for the C,M,K dot screens in colourseparation. In principle, any parameter combination in which no significant moiré isvisible is a solution to the problem, including the singular states at the centers of theforbidden zones (unless such a singular point is also covered by a forbidden zonebelonging to another moiré). As we have already seen in Sec. 10, the traditionalcombination of α=β=30° and qMK=qCK=1 is an example of such a singular point. But

when the accuracy of the printing process is not enough to guarantee the high precisionrequired for the use of a singular screen combination, it may be preferable to use astable moiré-free solution, i.e. a screen combination which is located in the 4D space atthe center of a moiré-free zone, with as high as possible tolerances in each of the 4parameters involved. Therefore, rather than looking for singular points (in the center ofthe forbidden zones), as proposed in [14], we will be looking for stable solutions in thecenter of the moiré-free zones.

Before we proceed to describe the algorithm itself, a word about the useful ranges ofthe 4 parameters is in order. It is clear from Fig. 15 that the angles α and β are limited

to the range 0° ≤ α+β ≤ 90°, but since we know experimentally that any angle

difference lower than 20° inevitably generates strong {1,1}-moirés (such as in Fig. 9(a))we may limit ourselves only to cases in which all of the 3 angles, α , β and γ, are larger

than 20°. This implies the following angle range (see Fig. 18): 20° ≤ α ≤ 50°,

20° ≤ β ≤ 70°–α. As for the frequency ratios, since we suppose that fC,fM ≤ fK it followsthat the useful ranges for qMK and qCK are: qCK,qMK ≤ 1. But since it is inappropriate to mix

within the same color image C,M,K screens of very different periods or frequencies wewill restrict ourselves to the following period ranges: TK ≤ TM,TC ≤ 1.5*TK, which meansin terms of frequency ratios: 2/3 ≤ qCK,qMK ≤ 1.

The method we adopted for finding moiré-free combinations of screens for colourseparation is a straightforward search, which runs with the specified steps through allthe parameter combinations, within their admitted ranges. It first searches for goodcombinations for the C,M,K screens, and only then, for each good solution it attemptsto find appropriate parameters for the Y screen. Here are the main steps of thealgorithm:

Step 1: Generating a list of the “dangerous” impulses in the spectrum of thesuperposition.

The aim of this step is to select from the list of impulses generated in the spectralconvolution of the C,M,K screens a relatively small subset which contains only the

26

“potentially dangerous” impulses. This will be further explained below. It shouldbe noted that what is needed here is not the actual impulse locations or amplitudes,but just their indices, such as: (2,0,-1,0,0,1).

Step 2: Scanning the 4D parameter space for moiré-free combinations.

In this stage we run through all combinations of the 4 parameters α , β, qMK, qCK

(within their admitted ranges, as described above). The step size should be smallenough to ensure that no moiré-free zones escape through the mesh, but not toosmall either in order to avoid excessive running time. For each parametercombination the program runs through the list of “dangerous” impulses andcalculates for each of them its actual coordinates in the spectrum, using Eq. (20). Ifany of the impulses fall inside the corresponding visibility circle, the currentparameter combination is rejected. Only if all the dangerous impulses fall outsidetheir visibility circle is the parameter combination accepted and registered in a filewhich accumulates the potentially good moiré-free combinations.

Step 3: Finding the best solutions inside the detected moiré-free zones.

Having completed the scanning of all the 4 parameter combinations in step 2, weare left with a file containing all the potentially good combinations found. As wehave seen above, these points are located within the moiré-free zones in the 4Dparameter space. In this step we search for the best solutions among these points,according to the tolerances they offer. Then each of the best solutions found mustbe experimented on high resolution films, in order to exclude the possibleexistance of residual moirés which are still visible but have not been detected bythe program (for instance, higher order moirés).

Step 4: Finding for each good C,M,K combination an appropriate Y screen.

After the proposed moiré-free combinations for the C,M,K screens have beentested and verified as good solutions, a similar approach is taken to find for eachof them a good Y screen. In this case the values of α , β, qMK and qCK are already

known, and the varying parameters are the angle δ and the frequency ratio qYK of

the Y screen. (Obviously, the list of “dangerous impulses” as well as Eq. (20)must be adapted to the case of 4 superposed screens).

The reason why the best Y screen is found separately is double: first, the introductionof a fourth screen in steps 1–3 would significantly increase the number of moirés whichoccur between the different layers, and make it practically impossible to find goodmoiré-free combinations between all 4 layers simultaneously. As a compromise, it isbetter to find the best possible moiré-free combinations for the C,M,K screens first,even at the expense of having somewhat stronger moirés with the Y screen (which are

27

anyway much less visible due to the light nature of the Y colour). As a second benefit,the separation of the Y screen from the others reduces from 6 to 4 the number ofdimensions to be scanned in step 2; this significantly reduces the complexity of theprogram, particularly in terms of running time.

Let us return now in more detail to step 1, concerning the number of impulsesinvolved. Since each of the C,M,K screens contributes 2 perpendicular Dirac-combs tothe convolution, the total number of impulses in the convolution, if only N impulses(harmonics) are considered from each side of the DC, is (2N+1)6. For N=2 this gives56=15625 impulses, and for N=3 the impulse number becomes as high as 76=117649.Fortunately, however, the actual number of impulses to be considered can besignificantly reduced by means of 3 different considerations:

First, according to our practical experience, an impulse generated by a harmonichigher than 2 very rarely contributes a significant visible moiré. It is therefore moreefficient to search for good solutions taking into account only 2 harmonics, and onlythen test each of the received solutions for possible higher order moirés.

Second, since we only deal with real images having no imaginary part, each impulsein the spectrum is accompanied by a twin impulse, symmetric with respect to the DC(see Sec. 2). Judicious consideration enables us to leave in the impulse list only oneimpulse from each such pair, thus reducing by half the number of impulses in the list.

Third, even after the above cut-downs in the number of impulses, most of theimpulses remaining in the list have no practical significance from the moiré point ofview since they represent moirés which are far too weak to be perceived, even whenthey appear at low frequencies (for instance, such are all the impulses having more than2 indices with the value of ±2). After discarding all these negligible impulses, we areleft with a list of only about 2000 “potentially dangerous” impulses, i.e. impulses whichmay represent, when they fall inside the visibility circle, significant moirés that shouldbe rejected.

However, as we already know, even these 2000 impulses are not all of the sameimportance. For example (see Fig. 9), a (1,0,-1,0,0,0)-moiré is much stronger than a(1,2,-2,-1,0,0)-moiré, and consequently its cutoff frequency (i.e. the frequency at whichit becomes invisible to the eye) is higher. In fact, strong moirés may still cause a roughand clearly visible linen-like structure even at very small periods, where the weakermoirés already disappear completely. Therefore the 2000 impulses in the list have beenclassified into several categories, according to the relative strength of their moirés. Eachcategory has been assigned its own visibility circle, according to its cutoff frequency.The cutoff frequency for each type of moiré has been found experimentally bymeasurements performed on a light-table, using various high-resolution films withdifferent screen frequencies, cell shapes and gray levels. In each case the observer

28

looked at the corresponding screen combination from a normal reading distance, and herotated the films slightly so as to decrease the period of the moiré in question (i.e. toincrease its frequency) until, at a certain moment, the moiré completely disappeared.The highest visible frequency of the moiré, just before its disappearance, was measuredand taken as its cutoff frequency (allowing for some extra safety margins). Moiré typeswith similar cutoff frequencies have been grouped into the same category. In ourspecific implementation we had 4 impulse categories: one for the strongest moirés (ofthe {1,1} type), a second one for the slightly weaker moirés of the {1,1,1} type, a thirdcategory for the medium-strength moirés such as {1,1,2} and {1,2,2}, and the lastcategory for all the weaker cases still in the list. It is interesting to mention that if noclassification of the impulses is carried out and all of them use the visibility circle of thestrongest moirés, the forbidden zones they generate in the 4D parameter space becomelarger and cover the whole of the space, so that no solutions can be found.

Note that these experimental measurements of the cutoff frequency have theadvantage of being performed using real halftone screen superpositions, thus taking intoaccount the irregular background and the masking effects which influence the moirévisibility. This is much better than using standard data taken from literature, which havebeen measured on ideal sinusoidal images, free of noise such as rosettes or textures dueto other moirés in the background.

13. Results and discussion

An important advantage of the above moiré minimization method is that, due to therelative simplicity of the calculations involved, it is capable of scanning the fulladmitted ranges of the 4 parameters. This provides a global, panoramic view of thewhole 4D parameter space, including the forbidden zones inside it and the moiré-freezones between them. Note that every large enough moiré-free zone should be detectedby this scan; moiré-free zones which “escape” through the scanning mesh (the samplingpoints in the 4D space) are inevitably smaller than the scanning step, which means thattheir tolerance margins are very low so that anyway they do not correspond to ourrequirements. (Such are, for example, all the singular points). Obviously, after animportant moiré-free zone is discovered, the program may be used again to rescan theinteresting zone alone in greater detail (“zoom in”).

The results obtained by this method are quite interesting. Several good screencombinations have been found, which gave in experimental tests a very satisfactory,uniform appearance with no evident apparition of moirés or other disturbing structures(some of these solutions correspond to previously known cases listed in [15]). Anexample of such a good solution is shown in Fig. 19. Note that each solution in factrepresents a whole family of equivalent screen combinations which can be receivedfrom each other by various symmetries and transformations (such as mirror-images,

29

rotations, scalings, flipping the colour labels between the 3 screens, etc...). Thetolerance obtained around a typical good solution is in the order of at least ±0.5° for theangles α and β and ±1% for the screen periods or frequencies (these values are

indicated for the narrowest dimension of the moiré-free zone; in its more prolongeddirections the tolerances may be even higher). This is significantly better than thetolerance required by the traditionally used singular screen combination of α=β=30°and qMK=qCK=1 (or any ather singular case), which is ±3´ (minutes of arc) for the angles

(DIN No. 16547). And moreover, since our good solutions are not singular states, evenwhen the specified tolerances are somewhat exceeded, the visual effect only consists ofweak, small-period moirés which gradually start to appear, while in singular moiré-freecases the slightest deviations immediately bring back from infinity large-period, highlyobjectionable moirés. In other words: our solutions are stable moiré-free states ratherthan unstable moiré-free states.

Another interesting result observed is that the best solutions in the moiré-free zones,according to our criteria of maximal tolerance, were systematically located on asymmetry axis of the 4D parameter space, such as: α=β, qMK=qCK (or any of its

symmetric counterparts, like: β=γ, qCK=qCM etc.). Perhaps this is not so surprising, after

all, since there is no a-priori reason why one side of the symmetry axis should bepreferred to the other. But the interesting thing is that these highly symmetric caseswere not necessarily the winners during the experimental tests, and often the casesfound most suitable for practical use were slightly off the symmetry axes. The reasonfor this result appears to be the following: in highly symmetrical cases the dots of the 3screens form together in the superposition a fine uniform geometrically ordered micro-structure which appears very regular to the eye. Now, even very weak moirés (includingmoirés due to higher harmonic orders!), which are normally completely negligible onthe irregular, noisy background of the screen superposition (even if their period isrelatively large), may become quite disturbing here. Even the slightest perturbation on auniform, geometrically ordered background is very clearly perceived by the human eyewhich is very sensitive to such periodic irregularities on a uniform background.

Therefore, even if the most promising cases obtained by the program are located on asymmetry axis, it might be adviseable to prefer a neighbouring case slightly off thataxis, in order to break the symmetry of the micro-structure generated in thesuperposition.

14. Conclusions

In this article a spectral approach for the analysis of superposition moiré effects ispresented. Using this approach, the development of the moiré analysis is systematicallysurveyed starting from the simplest, cosinusoidal grating case, through binary gratings

30

and square grids to the case of dot screen superposition, and then further to the case ofsuperposition moirés in colour separation. This is done based on a useful notationalformalism introduced for the identification, classification and labelling of the moiréeffects. Interesting results which have been demonstrated include the fact that theangles and periods of the moiré patterns are determined only by the angles and periodsof the superposed structures, while the intensity and the profile shape of the moirépatterns are basically determined by the opening ratios or the dot shape of thesuperposed structures. We also define the visibility circle for each type of moiré, theradius of which depends on several factors, including the perceptual contrast of themoiré.

Then, based on this background, in the second part of the article we analyze the moiréphenomenon from a different angle — the dynamic point of view. We introduce themoiré parameter space and show that the good parameter combinations are located in itwithin isolated moiré-free zones. This leads us to an algorithm for finding stable moiré-free screen combinations for colour separation, which have much better tolerances thansingular moiré-free screen combinations.

Acknowledgement

This research was partially supported by the Swiss National Fund (grant No. 21-31136.91).

31

References:

1. G. Oster, The Science of Moiré Patterns. Edmund Scientific Co., USA (second edition, 1969).

2. P. S. Theocaris, Moiré Fringes in Strain Analysis. Pergamon Press, UK (1969).

3. H. Takasaki, “Moiré topography,” Applied Optics, 9(6), 1467–1472 (1970).

4. Y. Nishijima and G. Oster, “Moiré patterns: their application to refractive index and refractive index

gradient measurements,” Jour. of the Optical Society of America, 5 4(1), 1–5 (1964).

5. J. A. C. Yule, “Moiré Patterns and Screen Angles,” Chap. 13 in Principles of Color Reproduction, pp.

328-345, John Wiley & Sons, Inc., New York (1967).

6. I. Amidror, “The moiré phenomenon in colour separation,” in Raster Imaging and Digital Typography

II , Proc. of the 2nd International Conference on Raster Imaging and Digital Typography, R. A.

Morris, J. André, Eds., pp. 98–119, Cambridge University Press (1991).

7. G. Oster, M. Wasserman and C. Zwerling, “Theoretical interpretation of moiré patterns,” Jour. of the

Optical Society of America, 54(2), 169–175 (1964).

8. A. Rosenfeld and A. C. Kak, Digital Picture Processing, Vol.1. Academic Press, Inc., Florida, USA

(second edition, 1982).

9. R. N. Bracewell, The Fourier Transform and its Applications. McGraw-Hill Publishing Company,

Reading, N.Y. (second edition, 1986).

10. R. W. G. Hunt, The Reproduction of Colour in Photography, Printing and Television. Fountain Press,

England (fourth edition, 1987).

11. O. Bryngdahl, “Moiré and higher grating harmonics,” Jour. of the Optical Society of America, 65(6),

685–694 (1975).

12. W. K. Pratt, Digital Image Processing, John Wiley & Sons, Inc., New York (second edition, 1991).

13. W. F. Schreiber, Fundamentals of Electronic Imaging Systems. Springer-Verlag, Berlin (1986).

14. P. Fink, PostScript Screening: Adobe Accurate Screens, Adobe Press, USA (1992); Chapters 9-11

and Appendix A, pp. 107-146.

15. J. Schoppmeyer, “Screen systems for multicolor printing”, U.S. Patent 4537470(1985).

32

u

v

f

–f

Amplitude

•

•

Impulse

Geometriclocation

Frequency vector

θθ

Fig. 1: The geometric location and amplitude of impulses in the 2D spectrum. To eachimpulse is attached its frequency vector, which points at the geometric location of theimpulse in the spectrum plane (u,v).

33

Fig. 2: (a) and (b) Cosinusoidal gratings and (c) their superposition in the imagedomain. (d) and (e) Top views of the respective spectra and (f) their convolution. Blackdots in the spectra indicate the geometric location of the impulses; the line segmentsconnecting them have been added only to clarify the geometric relations. (g), (h) and (i)Side views of the same spectra, showing the impulse amplitudes.

u

v

uu

v v

• •••••

• •• • •

••

••

f1

f2 f2f1+

f2f1–f1–

f2– f2–

f2

f1

f1–

f2f1––

f2 f1–

(d) (e) (f)

(a) (b) (c)

u

1/2

1/41/4

u

1/2

1/41/4

(g) (h) (i)

u

1/4

1/161/8 1/8

1/16 1/16

•

y

x

y

x

y

x

34

Fig. 3: (a) A third cosinusoidal grating and (b) its superposition with Fig. 2(c), in theimage domain; (c) and (d) a top view of their respective spectra in the spectral domain.The strongest visible moirés in (b), which are best seen from a distance of about 3 m,belong to the two impulse pairs marked by arrows in the spectrum (d).

u

v

f2f1–

f2–

f2

f1

f1–

f2 f1– f3f1–

f3 f1–

(d)(c)

v

f3

f3–•

•• u •

••

••

•••

••

••

••

• ••

•••••••

• ••

(a) (b)

35

Fig. 4: A symmetric square wave with period T and opening τ and its Fouriertransform. The dotted line indicates the envelope of the impulse train.

36

Fig. 5: (a) and (b) Binary gratings and (c) their superposition in the image domain;their respective spectra are the infinite combs shown in (d) and (e) and their convolution(f). (The scale in the image domain was reduced for the sake of clarity). The circle inthe center of spectrum (f) represents the visibility circle. It contains the impulse pairwhose frequency vectors are f 1– f2 and f2 – f1; this is the fundamental impulse pair of themoiré seen in (c). The dotted line in (f) shows the infinite comb of impulses thatrepresents the moiré.

x

y

x

y

x

y

(a) (b) (c)

(d) (e) (f)

v

u

v

f1f1–• •• ••

2f12f1–

••

••

•••

••

••

••

••

••

••

••

••

••

••

••••

••

•

••

••

••

••

••

••

••

••

•

••

••

••

•

••

••

v

•

•

••

•

•

•

•

•

••

•

••

•

••

••

•

•

•

•

••

u

••

•

•

•

•

•

•

•

•

•

•

f2

f2–

•

••

u

••

u

f22

f22–

f1 f2–f2 f1–

37

Fig. 6: (a) and (b) Binary gratings as in Fig. 5 but with (b) having half the frequency,and (c) their superposition; (d), (e) and (f) their respective spectra. (The scale in theimage domain was reduced for the sake of clarity). The visibility circle in the center ofthe spectrum (f) contains the impulse pair with frequency vectors f 1 – 2f2 and 2f2 – f1,which originate from the second harmonic of f 2, and represent the fundamental impulsepair of the moiré. Note that the moiré seen in (c) is a (1,-2)-moiré, but it still has thesame angle and frequency as the (1,-1)-moiré of Fig. 5, and only its intensity is weaker.

x

y

x

y

x

y

(d) (e) (f)

(a) (b) (c)

•

••

••

•••

••

••

•••

••

••

••

••

••

••

•

••

••

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••

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•

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•

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•

•• ••

•

•

••

•

•

••

•

•

••

•

•

••

•••

•

•••

••

•

•

••

•

••

f22–

• u

v v

f1

f2

f1– f2–• • ••

•

••

••

••

f24

f24–

2f12f1–

•

•• •

••

•••

u

v

•

•

••

•

•

•

••

•

••

•

••

••

••

•

••

u

••

•

•

•

•

•

•

•

u

•

f22 f23

f23–

•

•

f2 f1–2 f1 f2–2

38

Fig. 7: (a) Some of the different order moirés between two superposed gratings, and(b) enlarged view.

39

Fig. 8: A (1,-1)-moiré between two identical binary gratings. All three cases have thesame periods and angles, but their opening ratios are (a) 0.75, (b) 0.5, and (c) 0.25. (d),(e) and (f) The respective moiré profiles in terms of reflectance, as received from themathematical model. (g), (h) and (i) The same moiré profiles after their adaptation tothe human visual perception, i.e., in terms of density.

Reflectance

D

Reflectance

D

(b) (c)(a)

(e) (f)(d)

Reflectance

(h) (i)(g)

D

0

0.5

1

0.25

0.75

0

0.5

1

0.25

0.75

0

0.5

1

0.25

0.75

0

0.5

1

0.25

0.75

0

0.5

1

0.25

0.75

τ/T = 0.5τ/T = 0.75 τ/T = 0.25

0

0.5

1

0.25

0.75

40

Fig. 9: Three types of moirés between two square grids (left) or two dot-screens(right), which are generated by (a) two, (b) three, or (c) four of the gratings involved.

A

BC

D

A

B

C D

A

B

C D

41

Fig. 10: The spectral interpretation of the three types of moirés between two squaregrids shown in Fig. 9, which are generated by two, three or four of the gratingsinvolved. For the sake of clarity, only the frequency vectors of one of the twoperpendicularly symmetric moirés are shown in each case. The low frequency vectorialsum which corresponds to the fundamental impulse of the moiré in each of these casescan be graphically found by the parallelogram law (the dashed axes here denote therotated layers).

(a): (1,0,-1,0)-moiré (b): (1,1,-1,0)-moiré (c): (1,2,-2,-1)-moiré

f1

u

v

•

f2

The vectorialsum

f3–

α

f4

f1 f2 f3–+

u

v

•f1f3–

The vectorialsum

f2f4

f1 f3–

α u

v

f1

f3–2 f4–

f22

The vectorialsum

αf1 f22 f3–2 f4–+•

42

Fig. 11: (a) A square white dot d(x,y) with side τ ; (b) its continuous spectrum D(u,v) =τ 2 sinc(τu) sinc(τv); (c) a nailbed with period T and amplitude 1; (d) its spectrum — anailbed with period 1/T and amplitude 1/T 2; (e) a screen of square white dots — theconvolution of (a) and (c); and (f) the spectrum of the screen — the product of (b) and(d): a nailbed that samples the “envelope” (b) at intervals of 1/T, scaling its amplitudeby 1/T 2.

(a) (b)

(c) (d)

(e) (f)

T

τ

τ/2−τ/2

1

1 = white0 = black

τ

T

1.

..

..

.

..

.

−2/τ−1/τ

0

1/τ

2/τ−2/τ

−1/τ

0

1/τ

2/τ0

τ

−2/τ−1/τ

0

1/τ

2/τ−2/τ

−1/τ

0

1/τ

2/τ0

τ2

0

00

0

00

1/Τ2

−2/τ−1/τ

0

1/τ

2/τ−2/τ

−1/τ

0

1/τ

2/τ

0

−2/τ−1/τ

0

1/τ

2/τ−2/τ

−1/τ

0

1/τ

2/τ

0

τ /Τ2 2

43

Fig. 12: The influence of the size and shape of the screen dots on the moiré betweentwo superposed screens. Only the size and the shape of the “spot function” of the moiréare affected, but not the period and the direction of the moiré. In both images twosimilar screens, with identical frequencies and gradually increasing dots, are superposedat an angle difference of 2°. (a) The superposition of two screens with increasinglylarge circular dots gives circular moiré spots of increasing size. (b) The superposition oftwo screens with increasingly large square dots give square moiré spots of increasingsize.

Fig. 13: The influence of the dot size and of the dot shape of the superposed screenson the moiré intensity. Both images show the superposition of two screens, one at 28.3°and with frequency f1 and the other at 51.34° and with frequency f2 = 1.153 f1. In bothimages the gray level of the first screen is varied vertically from 0 to 1 (by modifyingthe dot size within the fixed screen period), and the gray level of the second screenvaries horizontally from 0 to 1. This means that each image contains all the possible dotsize combinations of the two screens. The only difference between images (a) and (b) isin the shape of the screen dots: the dots are circular in (a), and rhombic (i.e., squaresrotated by 45°) in (b). In each of the images two distinct moirés appear: The largesquare pattern at –10.4°, which belongs to a {1,2,2}-moiré, and the small square patternat 20.6°, which is a {1,1}-moiré. Note that the intensity of each of the two moiréswithin each of the images varies with the gray-level combination of the two screens,and reaches its maxima in different gray-level combinations. The influence of the dotshape is demonstrated by the significant difference in the moiré intensities between thetwo images.

44

Fig. 14: A panoramic view of the 2D parameter space (α ,q) within the range0° ≤ α ≤ 45°, 1 ≤ q ≤ 2. It shows the most significant moirés between two superposedregular screens in this range, and illustrates how their period varies as a function of theangle difference α and the frequency ratio q between the two screens. Darker shadesrepresent bigger moiré periods. Each moiré is centered at a singular point in which itsperiod is infinitely large; this point is surrounded by a spherical zone in which theperiod of the moiré gradually decreases from the center outwards. The contour linearound each moiré delimits the parameter combinations (α,q) for which thefundamental impulse of the moiré is located in the spectrum inside the visibility circle.Note that the (1,0,-1,0)-moiré is clearly more dominant than the others. Note also thatthe (2,2 ,-2,-1)-moiré is already too weak to be visible; in practice it is assimilated by theresidues of the strong (1,0,-1,0)-moiré.

10 2030

40

1

1.2

1.4

1.6

1.8

2

α

q

(1,0,-1,0)

(2,0,-1,0)

(1,1,-1,0)

(2,0,-1,1)

(2,1,-2,0)

(2,1,-1,0)

(1,2,-2,-1)

(2,2,-2,-1)

(1,2,-1,-1)

45

Fig. 15: A schematic view of the three superposed C,M,K screens (each representedby its two main axes) showing the conventional notations used in the text.

K

K

M

CM

C

αβ

α βγ

46

Fig. 16: A (0,1,-1,0,1,0)-moiré between three screens (a), and its vector diagram (b).By convention it is assumed that the K screen is fixed, and only the M and C screensare free to vary. For any choice of α and β there exists one set of qMK and qCK (i.e., oneset of f M and fC) for which fM+ f C falls on –f K, thus yielding a singular case (where thevectorial sum is 0). This means that the locus of all the singular points of this moiréforms a 2D manifold in the 4D space (α , β, qMK, qCK). Note that one of these singularpoints, the point with α = β = 30° and qMK = qCK = 1, is the case used in traditionalcolour printing.

K

K

M

CM

C

α β

fK

fC

fM

The vectorialsum

fM+fC

α β

(b)(a)

.fK–

K

M

C

47

Fig. 17: Two 2D sections through the 4D parameter space (α , β, qMK, qCK) showingtheir intersections with the singular manifold of the (0,1,-1,0,1,0)-moiré and theforbidden zone surrounding it. (a) A section along the diagonal plane α = β, qMK = qCK.(b) A section along the (α ,β) plane with qMK = qCK = 1. Darker shades represent highermoiré periods. Note that the singular point α = β = 30°, qMK = qCK = 1 is included in thissingular manifold.

Fig. 18: The admitted range for the angles α , β and γ. Any point outside the grayrange violates at least one of the conditions α ≥ 20°, β ≥ 20°, γ = 90° – (α + β) ≥ 20°.

10 2030

40 50

0.6

0.8

1

1.2

α10 20

3040 50

10

20

30

40

α

α

β

20°

20°

50°

50°β = 70°– α

30°

40°

30° 40°

γ = 20°γ = 30°γ = 40°50°

48

Fig. 19: An example of a stable moiré-free combination of three screens obtained bythe algorithm. Small, irregular rosettes, of about the same size as in the conventional30°-30°-30° singular (unstable) moiré-free case, can be seen in the superposition, but nosignificant moirés (such as in Fig. 9) are present.